Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by $G_x$ the stabilizer of $x$ and by $G_x^0$ the identity connected component of $G_x$. Does anyone know a reference where it it proved the the function $f(x)=\vert G_x/G_x^0\vert\cdot vol_d(G(x))$ is continuous
on $M$? Here, $\vert .\vert$ denotes the cardinality, and $vol_d$ the $d$-dimensional volume (induced by the restriction of the metric $g$ on the orbit).

How about a pseudo-Riemannian extension of the above? If $(M,g)$ is pseudo-Riemannian,
then one defines $f(x)$ as above when $G(x)$ is a nondegenerate submanifold, and $f(x)=0$ otherwise. Is such $f$ continuous?

Note that $f(x)\ne 0$ only if $G(x)$ has dimension $d$, i.e., if $G(x)$ is either a principal or an exceptional orbit. The function $v(x)=vol_d(x)$ is only continuous at points
$x$ whose orbit is principal, but it fails to be continuous at points with exceptional orbit.

1 Answer
1

To my understanding, Proposition 1 in this paper of Pacini, TAMS 2003 gives exactly the proof that you ask for in the Riemannian case; namely, that the volume of orbits: $$vol\colon M\to \mathbb R, \quad vol(x)=\int_{G(x)} i^*(vol_M),$$ where $i\colon G(x)\hookrightarrow M$ is the immersion of the $G$-orbit through $x$ and $vol_M$ is the volume form of $M$, is a continuous function on $M$, vanishing exactly at singular orbits.

it has a continuous extension $vol\colon M\to\mathbb R$ that is zero on the singular points $M^{sing}=M\setminus M^{reg}$;

$vol^2\colon M\to\mathbb R$ is smooth.

Note that Pacini defines the volume of an orbit not by using the volume of the image, but rather by integrating the pull-back of the volume form. These are the same thing only if the immersion is $1$-to-$1$ (e.g., for principal orbits $G/P$). For an exceptional orbit $G/K$, the immersion is $k$-to-$1$, where $k$ is the number of sheets on the covering by a principal orbit $G/P\to G/K$, so the volume of the image is multiplied by $k$. This correction factor is precisely the cardinality of the fiber, $k=|P/K|$, as pointed out by the OP.

Regarding the second question, adapting the proof to the nondegenerate semi-Riemannian case should be straightforward.

Edit: I recently realized that also the classic paper by Hsiang-Lawson, JDG 1971 (see first few lines of page 7) cites the continuity of the volume function above in $M$ (being zero on singular points) and smoothness in the set of regular points. Although they do not provide an explicit proof, they say it is straightforward from the Slice Theorem. There are also many nice examples following that.